# Topics in Geometry: Quantitative Geometry and Topology

## Overview

This course is an introduction to quantitative and asymptotic
techniques in geometry and topology. Quantitative geometry studies how
the size or complexity of a map or space affects its geometry and
topology, and how maps and spaces behave at large scales or under
various asymptotics. We will introduce some of the ideas and methods
of quantitative geometry and apply them to questions in areas like
geometric group theory and systolic geometry.

Tentative outline:

- Quantifying simple connectivity: the Dehn function
- Higher homotopy groups

## Basic information

- Instructor: Robert Young (ryoung@cims.nyu.edu)
- Office: CIWW 601
- Office hours: by appointment
- Lectures: CIWW 317, Tuesdays, 11:00--12:50

## Further reading

### Geometric group theory

Geometric group theory studies the geometry of Cayley graphs of groups
and other spaces on which groups act. Given a group \(G\), there are
typically many spaces on which \(G\) acts freely, cocompactly, and by
isometries. One of the main goals of geometric group theory is to find
invariants that describe how the geometry of these spaces depends on
\(G\). The Dehn function is one such invariant -- if \(G\) acts on
\(X\), then \(G\) and \(X\) have Dehn functions with the same growth rate.

### Nilpotent groups and subriemannian geometry

In the last few lectures, we've studied the Heisenberg group and its
scaling limit. More generally, the scaling limit of any nilpotent Lie
group is a subriemannian manifold. There are still many questions
about the geometry of surfaces in subriemannian manifolds and
isometries and quasi-isometries between them.

- Enrico Le Donne, "A
Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory
Spaces, and Regularity of Their Isometries"
- G. Baumslag, C. F. Miller III, and H. Short, "Isoperimetric
inequalities and the homology of groups"
- R. Young, "Filling
inequalities in nilpotent groups through approximations
- S. Wenger, "Nilpotent
groups without exactly polynomial Dehn function"
- Isenrich, Pallier, Tessera, "Cone-equivalent nilpotent
groups with different Dehn functions"

### Quantitative homotopy theory

One of my goals with this course was to talk about quantitative
homotopy theory and some recent work on bounding the growth of
homotopy classes. Unfortunately, we ran out of time, but here are some
of the references I was planning to use:

### Differentiability and rectifiability

In a different direction, the

notes
for

my previous
topics course cover quantitative ways of looking at the
differentiability of maps between spaces and the rectifiability of
subsets in a space.

## Notes

### Transcribed notes

Notes on quantitative topology (Lectures 1-7), transcribed by the class.

### Scanned notes